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Complement of an Event

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Hi. This tutorial covers the complement of an event.

All right. Let's start with the definition. The complement of an event is the set of all outcomes not in an event. OK. So if we have an event, the complement contains all the outcomes that weren't in that event.

All right. So let's take a look at two experiments. The first experiment is just to simply draw a card. So what we'll do is we'll call drawing a card-- or the event we'll call as event A. So we'll label A as the event. And the event is that the card shows a heart. OK. So event A-- card shows a heart.

So the complement of that event-- first of all, we notate it as A with a little tick mark there. So if the event is A, the complement is-- we write like this. So then what the complement is, then, is that the card shows a diamond, spade, or club. So anything that's not a heart belongs in the complement.

So, again, event A is a heart. Event A prime, or the complement of A, shows everything else. OK.

So the second example here is the experiment will be to flip a coin. So we flip a coin. It's either going to be heads or tails. So if the event, event B, is the event of the coin shows a head, shows heads, then the complement of the event is that the coin shows tails. So if the event is B, the complement is B prime. OK.

Now, the probability of the complement is always the probability of the event subtracted from 1. So if we want to calculate the probability of A prime, the complement, all we do is take 1 minus the probability of A. Then, really, if we just flip the formula around, the probability of A equals 1 minus the probability of the complement of A.

All right. So putting that into practice, if the probability of drawing a face card from a deck is about 0.23, the probability of drawing a non-face card-- so an ace through-- sorry-- ace through 10 is about 0.77. So the probability of the complement-- or the event plus the probability of the complement, they need to add to 1. So if the probability of A is 0.23, the probability of the complement of A is 1 minus 0.23, which is 0.77.

So sometimes this idea of a complement helps to calculate probabilities sometimes in a faster way. So this table here shows the probability of a person accumulating specific amounts of credit card charges over a 12-month period. So the question now is, what's the probability that a person's total charges during the period are $500 or more?

OK. So if we look at this probability distribution, we can see all of the different charge intervals and then the corresponding probabilities. So we're looking for the probability of 500 or more. So really, we're talking about all of these outcomes here-- so 500 all the way through the end.

So what we could do is we could add up-- 1, 2, 3, 4, 5, 6-- these six values. But what we could do instead is consider the complement of the event. So if the event is 500 or more, the complement would be basically 0 to 499.

So what we can do to find the probability of 500 or more is just to simply take 1 minus the probability of 0 to 499. So to calculate that, we could just do 1 minus-- and then we just add up 0.31 and 0.18. And if we add those two together, that's going to give me 0.49. So 1 minus 0.49 is 0.51. So there's a 51% chance that somebody will accumulate credit card charges of 500 or more.

And that 0.51 will be the same number you get if you add up those six values. But sometimes it's just going to be easier to consider the complement to calculate a probability quicker than determining the probability of the event itself. All right. So that has been your tutorial on the complement of an event. Thanks for watching.